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# Perfect square and Positive Integer | TOMATO B.Stat Objective 115

Try this TOMATO problem from I.S.I. B.Stat Entrance Objective Problem based on Perfect square and Positive Integer. You may use sequential hints.

Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Perfect square and Positive Integer.

## Perfect Square & Positive Integers ( B.Stat Objective Question )

If n is a positive integer such that 8n+1 is a perfect square, then

• n must be odd
• 2n cannot be a perfect square
• n must be a prime number
• none of these

### Key Concepts

Perfect square

Positive Integer

Primes

But try the problem first…

Answer: 2n cannot be a perfect square.

Source

B.Stat Objective Problem 115

Challenges and Thrills of Pre-College Mathematics by University Press

## Try with Hints

First hint

Let $$(8n+1)=k^{2}$$ be a perfect square so k is found to be odd here as 8n+1 is odd.

Second Hint

$$\Rightarrow 8n=k^{2}-1$$

$$\Rightarrow 8n=(k-1)(k+1)$$

$$\Rightarrow 2n=\frac{(k-1)(k+1)}{4}$$

Final Step

$$\Rightarrow (\frac{k-1}{2})(\frac{k+1}{2})$$

here (k-1) and (k+1) are consecutive even numbers then $$(\frac{k-1}{2})(\frac{k+1}{2})$$ are consecutive even numbers

$$2k \times (2k+2)= 2^2{k(k+1)}$$ is not a perfect square as product of two consecutive numbers proved here as not a perfect square

So, 2n is not perfect square.

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